Esby,
The tires are set at 13 psi here at the factory and that is where we level and set the decks. You can raise the tire pressure if you like but I would raise it in each machine if you do. That way you'll still maintain the cutting heights as close as possible.
Personally I like 16-18 psi. because I find I get better traction, less turf damage and the ride really doesn't suffer enough for me to notice.
The shake in the front caster wheel is a simple adjustment that should be done each year when you lubricate the front caster pivots. After you install the grease zerk and remove the caster pivot cap you want to turn the nylock nut on the caster pivot all the way down and the back it off a 1/4 turn. This sets the "pre-load" on the tapered roller bearing and prevents the "shopping care" shake you've described. Once you've done that, grease the caster pivot until you see fresh grease appear, wipe away the dirty grease with a rage then install the cap and replace the zerk with the threaded plug. That's all there is to it.
Please let us know if we can be of any further assistance.
Thanks
Terry